Solvable Models of Resonances and Decays

Abstract

Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models starting the classical Friedrichs result and carrying further to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlying resonance effects and at time evolution of the related unstable systems.

Spectral and resonance properties of the Smilansky Hamiltonian

“…We see that the behavior beyond the weak-coupling regime is different, the pole trajectories return to the real axis and they may even end up as an isolated eigenvalue. This is not surprising, however, since a similar behavior is known from other resonance models [8] being first observed in the classical Lee-Friedrichs model [12].…”

Can we trust the relationship between resonance poles and lifetimes?

“…In particular the decay is approximately exponential for short times but the the expected survival time, that is, the integralof the survival probabilitywith respect to time, diverges. See [6] and section 9.3 of [7] for examples of this phenomenon.…”

On the spectrum of Schrödinger-type operators on two dimensional lattices

“…In the current paper, as we consider the discrete spectrum above the essential spectrum, we are interested in high energy resonances. Among various definitions, we mainly follow to [1,3,34,38]: in the momentum representation, a resonance of energy e max , where e max is the top of the essential spectrum, is a nonzero solution f of the eigenvalue equation H a,b (µ)f = e max f belonging to L 1 (T 2 ) \ L 2 (T 2 ) ; see also [6,7,11,22] and the references therein for other notions of resonances. Using the momentum representation, in Theorem 3.5 below we completely characterize the threshold eigenfunctions and threshold resonances, also finding them explicitly.…”